This work deals with the numerical solution of systems of oscillatory second-order differential equations which often arise from the semi-discretization in space of partial differential equations. Since these differential equations exhibit pronounced or highly) oscillatory behavior, standard numerical methods are known to perform poorly. Our approach consists in directly discretizing the problem by means of Gautschi-type integrators based on $\operatorname{sinc}$ matrix functions. The novelty contained here is that of using a suitable rational approximation formula for the $\operatorname{sinc}$ matrix function to apply a rational Krylov-like approximation method with suitable choices of poles. In particular, we discuss the application of the whole strategy to a finite element discretization of the wave equation.

翻译：本文研究振荡型二阶微分方程系统的数值解法，此类方程通常源于偏微分方程的空间半离散化。由于这些微分方程呈现显著（或高度）振荡特性，标准数值方法的求解效果往往不佳。我们提出的方法通过采用基于$\operatorname{sinc}$矩阵函数的Gautschi型积分器直接离散化问题。本文的创新之处在于：利用$\operatorname{sinc}$矩阵函数的合适有理逼近公式，结合极点选择策略，采用有理Krylov类逼近方法进行求解。特别地，我们讨论了将该完整策略应用于波动方程有限元离散化的具体方案。